We use Nicholson's approximation — found in [1]} but see [2]} for a recent elementary proof. We state the slightly stronger form of Johansson [3]}. Given \(M > 0\) , there exists a constant \(C > 0\)  such that
\(|\theta ^{1/3} J_{2 \theta + x \theta ^{1/3}} (2 \theta ) | &\le C \exp \left(-\frac{|x|}{4} \min \left(\frac{1}{4} \theta ^{1/3}, \sqrt{|x|}\right) \right), \qquad \forall x > -M;  \\\lim _{\theta \rightarrow \infty } \theta ^{1/3} J_{2 \theta + x \theta ^{1/3}} (2 \theta ) &= Ai(x), \qquad \text{uniformly for\ } x \in [-M, M]. \) 
